I feel it is my duty to pose a situational challenge to this question.
You are entirely out of line:
You are paying more attention to your colleagues duties than your own.
You are eavesdropping on office conversations which are none of your business.
You are spending office hours concerning yourself with others' personal matters.
You are exposing your ...
While I don't disagree with most of the current answers, I think they all miss the crux of the question's 2 critical points:
It's never an employee's responsibility to tell off a coworker.
This is management's responsibility.
If you take it upon yourself to tell off your coworker, you're opening yourself to potential problems. At a minimum, you've made an ...
asking our supervisor to buy an XBox X Series console at a black Friday sale for him, at regular working hours.
I assume you mean, he's asking the supervisor to buy it with company funds, and get it for employees to use. What's wrong with that?
Companies often buy games for employees to use in breakrooms. Game consoles are especially common.
Would your ...
Ask yourself these three questions (context taken from your narrative):
Is being talkative a problem?
Are they disturbing you / others from getting their work done?
Are they speaking in loud voices?
Are they arguing all the time?
Are they disrupting the concentration and focus needed for work?
Is being friendly a problem?
Are they violating the personal ...
"I don't want us to trouble our Supervisor." Your Supervisor can speak up for himself. Could be that the supervisor asked, "what would help team morale? I have a budget from the company for this" and your co-worker said, the x-box. I probably would have asked for a hot mocha latte with whipped cream.
I'd leave it alone.
Is it my responsibility to tell a team member off whom I think is crossing the line
From what you describe here, it's hard to say if your team member's behavior is appropriate or not. For the sake of discussion let's assume that it is. Unfortunately people who behave inappropriately have little of the self-awareness necessary to listen to or take advice....
Whether something qualifies as (workplace) bullying can often be very subjective. My take is that actual bullying is most often repetitive and akin to assault and should not be confused with people who are merely socially inept.
You mention many examples, which seems to indicate that you would like to address each situation in some unique way. I find that ...
The managment told me recently that they would give the rest of the team a new task and they want me to not get involved into it in order to see how the team is functioning without me.
This sounds like they want to see if there are any gaps in the team's ability to handle projects without you. It's good business to avoid having any business functions with ...
How to react if the team accepts you as leader but the managment tells you to back off?
If you were in fact the official leader of your team, you would expect your teammates to respect your decisions. They may not always like your decisions, but they should feel free to discuss them with you, and understand that leaders bear the ultimate responsibility.